Finite Group Databasebeta

← #10800523

Subgroups — #10800523

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3455C211yesmaximal, characteristic
2Dic3×#117C211yesmaximal, characteristic, complemented
3#82209C211yesmaximal, characteristic, complemented
4#71840C211yesmaximal, characteristic, complemented
5#71839C211yesmaximal, characteristic, complemented
6#71816C211yesmaximal, characteristic, complemented
7#71815C211yesmaximal, characteristic, complemented
8#71712C212yesmaximal, complemented
9#71711C212yesmaximal, complemented
10#71643C212yesmaximal, complemented
11C4×#837C212yesmaximal, complemented
12#21419—31nomaximal, Sylow
13C3×#542C414yescomplemented
14C3×#541C412yescomplemented
15C3×#540C412yescomplemented
16C3×#539—22no—
17C3×#538—22no—
18C3×#533C2211yescharacteristic
19C3×#532C2211yescharacteristic
20C3×#529—24no—
21C3×#525C2212yes—
22C3×C4×#41C2212yes—
23C2×C3×#117C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C22×C4×Dic3C2211yescharacteristic
26C2×#837C2212yes—
27C2×#837C2212yes—
28C2×#837C2212yes—
29C2×#837C2212yes—
30#5207C2212yes—
31#5000—24no—
32Dic3×C4⋊C4—22no—
33Dic3×C22⋊C4—22no—
34#4932C2211yescharacteristic
35#4932C2211yescharacteristic
36C42×Dic3C2211yescharacteristic
37C42×Dic3C2211yescharacteristic
38#4924C2211yescharacteristic
39#4924C2211yescharacteristic
40C4×#208C2212yes—
41#4532C2212yes—
42#4532C2212yes—
43#4532C2212yes—
44#4532C2212yes—
45#3455D311yescharacteristic
46#2921—31no—
47C4×#117—31no—
48#1995—31no—
49#1994—31no—
50#1963—31no—
51#1962—31no—
52#1897—32no—
53#1896—32no—
54#1759—32no—
55C4×#104—32no—
56C2×C3×#41C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C3×C4⋊Q8C2×C411yescharacteristic, complemented
60C3×C4⋊Q8C2×C412yescomplemented
61C3×#127C2×C411yescharacteristic, complemented
62C3×#126—24no—
63C3×#126—24no—
64C3×#125—24no—
65C3×#125C2×C412yescomplemented
66C3×#124—24no—
67C3×#124C2×C412yescomplemented
68C3×#123—22no—
69C3×#123—24no—
70C3×#123—24no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8C2×C414yes—
74C3×#121—22no—
75C3×#121—24no—
76C3×#121C2×C414yes—
77C3×#120—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8C2×C414yescomplemented
80C3×C4×D4—24no—
81C3×C4×D4—24no—
82C3×C4×D4—22no—
83C3×C4×D4C2×C414yescomplemented
84C3×#117—22no—
85C3×#117C2×C412yes—
86C3×#117C2×C412yes—
87C3×#117D414yes—
88C3×#117C2312yes—
89C2×C3×C4⋊C4D412yes—
90C2×C3×C22⋊C4D412yes—
91C2×C3×C42—22no—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95#926—22no—
96#926—22no—
97C2×C4⋊Dic3—21no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3—22no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C2×#208C2312yes—
108C2×#208C2312yes—
109#837—24no—
110#837—24no—
111#837—24no—
112#837—24no—
113C3⋊#95—22no—
114C3⋊#95—22no—
115#830—24no—
116C4⋊(C3⋊C8)—22no—
117C4⋊(C3⋊C8)—22no—
118#803—22no—
119C4×C3⋊C8—22no—
120#542Dic314yescomplemented
121#541Dic312yescomplemented
122#540Dic312yescomplemented
123#539—22no—
124#538—22no—
125#533C2×D311yescharacteristic
126#532C2×D311yescharacteristic
127#529—24no—
128#525C2×D312yes—
129C4×#41C2×D312yes—
130C2×#117—31no—
131C2×#117C2×D311yescharacteristic
132C22×C42—31no—
133#433—64no—
134#423—31no—
135#423—31no—
136C2×#104—32no—
137C2×#104—32no—
138C2×#104—32no—
139C2×#104—32no—
140#407—32no—
141C4×M16—32no—
142C4×C4⋊C4—62no—
143C4×C22⋊C4—62no—
144#376—31no—
145#376—31no—
146C43—31no—
147C43—31no—
148#339—32no—
149#339—32no—
150#339—32no—
151#339—32no—
152C24×C3C2×D411yescharacteristic
153C3×#41—22no—
154C3×#41—22no—
155C3×#41—22no—
156C3×#41—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8C22×C412yes—
159C2×C3×D4—22no—
160C2×C3×D4—24no—
161C2×C3×D4—22no—
162C2×C3×D4C22×C412yes—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4C22×C412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—21no—
176C22×Dic3—24no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—44no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4C22⋊C414yes—
187C3×C4⋊C4C22⋊C414yes—
188C3×C4⋊C4C22×C414yes—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4C22⋊C414yes—
195C3×C22⋊C4C22⋊C414yes—
196C3×C42—24no—
197C3×C42—22no—
198C3×C42C22×C412yes—
199C3×C42C22×C411yescharacteristic
200C3×C42C22×C411yescharacteristic
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203#217—24no—
204C4⋊Dic3—24no—
205C4×Dic3—24no—
206C4×Dic3—21no—
207C4×Dic3—21no—
208C4×Dic3—21no—
209C4×Dic3—21no—
210C4×Dic3—44no—
211C4×Dic3—44no—
212C4×Dic3#4112yes—
213C4×Dic3#4112yes—
214C4×Dic3#4112yes—
215C4×Dic3#4112yes—
216#208—22no—
217#208—22no—
218#208—22no—
219#208—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×#41C22×D312yes—
223C23×C4—61no—
224C23×C4C22×D311yescharacteristic
225C2×M16—32no—
226C2×M16—32no—
227C4⋊Q8C2×Dic311yescharacteristic, complemented
228C4⋊Q8C2×Dic312yescomplemented
229#127C2×Dic311yescharacteristic, complemented
230#126—24no—
231#126—24no—
232#125—24no—
233#125C2×Dic312yescomplemented
234#124—24no—
235#124C2×Dic312yescomplemented
236#123—22no—
237#123—24no—
238#123—24no—
239C22⋊Q8—22no—
240C22⋊Q8—24no—
241C22⋊Q8C2×Dic314yes—
242#121—22no—
243#121—24no—
244#121C2×Dic314yes—
245#120—22no—
246C4×Q8—22no—
247C4×Q8C2×Dic314yescomplemented
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—22no—
251C4×D4C2×Dic314yescomplemented
252#117—22no—
253#117—62no—
254#117—62no—
255#117C2×Dic312yes—
256#117C2×Dic312yes—
257#117#6714yes—
258#117C22×D312yes—
259C2×C4⋊C4—61no—
260C2×C4⋊C4#6712yes—
261C2×C22⋊C4—61no—
262C2×C22⋊C4#6712yes—
263C2×C42—22no—
264C2×C42—31no—
265C2×C42—31no—
266C2×C42—31no—
267C2×C42—31no—
268C2×C42—62no—
269C2×C42—62no—
270C2×C42—61no—
271C2×C42—62no—
272C2×C42—62no—
273C2×C42C22×D311yescharacteristic
274C2×C42C22×D311yescharacteristic
275C4⋊C8—62no—
276C4⋊C8—62no—
277#104—64no—
278#104—64no—
279#104—64no—
280#104—64no—
281C22⋊C8—64no—
282#97—62no—
283C4×C8—62no—
284#95—62no—
285#95—62no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—22no—
289C23×C3—42no—
290C23×C3C2×C22⋊C412yes—
291C23×C3C22×D411yescharacteristic
292C3×Q8—22no—
293C3×Q8—22no—
294C3×Q8—82no—
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—82no—
300C3×D4—84no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—84no—
314C2×C3×C4C2×C22⋊C414yes—
315C2×C3×C4C2×C22⋊C412yes—
316C2×C3×C4C2×C22⋊C412yes—
317C2×C3×C4C2×C22⋊C412yes—
318C2×C3×C4C23×C411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×Dic3—24no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—48no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C3⋊C8—42no—
334C3⋊C8—42no—
335C24C2×#6711yescharacteristic
336#41—22no—
337#41—22no—
338#41—22no—
339#41—22no—
340C2×Q8—22no—
341C2×Q8C22×Dic312yes—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4C22×Dic312yes—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—61no—
353C22×C4—64no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—124no—
358C22×C4C22×Dic312yes—
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6712yes—
361C22×C4C2×#6712yes—
362C22×C4C2×#6712yes—
363C22×C4C23×D311yescharacteristic
364M16—62no—
365M16—62no—
366M16—62no—
367M16—62no—
368C2×C8—62no—
369C2×C8—62no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—64no—
376C4⋊C4#21714yes—
377C4⋊C4#21714yes—
378C4⋊C4C22×Dic314yes—
379C22⋊C4—24no—
380C22⋊C4—22no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—64no—
385C22⋊C4#21714yes—
386C22⋊C4#21714yes—
387C42—24no—
388C42—22no—
389C42—32no—
390C42—32no—
391C42—32no—
392C42—32no—
393C42—64no—
394C42—61no—
395C42—61no—
396C42—61no—
397C42—61no—
398C42—124no—
399C42—124no—
400C42C22×Dic312yes—
401C42C22×Dic311yescharacteristic
402C42C22×Dic311yescharacteristic
403C42C2×#6712yes—
404C42C2×#6712yes—
405C22×C3—22no—
406C22×C3—24no—
407C22×C3—22no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—44no—
411C22×C3—42no—
412C22×C3C22×C22⋊C411yescharacteristic
413C22×C3#52511yescharacteristic
414C22×C3#52511yescharacteristic
415C3×C4—24no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—44no—
420C3×C4—42no—
421C3×C4C4×#4111yescharacteristic
422C3×C4C4×#4111yescharacteristic
423C3×C4#53211yescharacteristic
424C3×C4#53211yescharacteristic
425Dic3—24no—
426Dic3—42no—
427Dic3—42no—
428Dic3—42no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—42no—
433C23C2×#21712yes—
434C23C22×#6711yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—82no—
438D4—22no—
439D4—22no—
440D4—42no—
441D4—42no—
442D4—82no—
443D4—84no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—64no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—84no—
462C2×C4—128no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4C2×#21714yes—
467C2×C4C2×#21712yes—
468C2×C4C2×#21712yes—
469C2×C4C2×#21712yes—
470C2×C4D3×#4111yescharacteristic
471C2×C4D3×#4111yescharacteristic
472C2×C4D3×#4111yescharacteristic
473C2×C4D3×#4111yescharacteristic
474C2×C4C23×Dic311yescharacteristic
475C2×C4C22×#6711yescharacteristic
476C8—122no—
477C8—122no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—42no—
481C2×C3—82no—
482C2×C3#229711yescharacteristic
483C2×C3#290112yes—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—44no—
491C22#582811yescharacteristic
492C22#582811yescharacteristic
493C22C22×#21711yescharacteristic
494C4—22no—
495C4—24no—
496C4—22no—
497C4—22no—
498C4—42no—
499C4—44no—
500C4—64no—
501C4—122no—
502C4—122no—
503C4—122no—
504C4Dic3×#4111yescharacteristic
505C4Dic3×#4111yescharacteristic
506C4#583211yescharacteristic
507C4#583211yescharacteristic
508C3#2141911yescharacteristic, complemented, Sylow
509C2—21no—
510C2—21no—
511C2—42no—
512C2—82no—
513C2#8408611yescharacteristic
514C2#8412012yes—
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3455C2
socleC22×C3C22×C22⋊C4
cosocleC22×C3×C4C24
hypercenter#3455D3
hypocenterC3#21419
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×#117 ◁ #3455

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. Dic3×#117C2semidirect
3. #82209C2semidirect
4. #71840C2semidirect
5. #71839C2semidirect
6. #71816C2semidirect
7. #71815C2semidirect
8. #71712C2semidirect
9. #71711C2semidirect
10. #71643C2semidirect
11. C4×#837C2semidirect
13. C3×#542C4semidirect
14. C3×#541C4semidirect
15. C3×#540C4semidirect
59. C3×C4⋊Q8C2×C4semidirect
60. C3×C4⋊Q8C2×C4semidirect
61. C3×#127C2×C4semidirect
65. C3×#125C2×C4semidirect
67. C3×#124C2×C4semidirect
79. C3×C4×Q8C2×C4semidirect
83. C3×C4×D4C2×C4semidirect
120. #542Dic3semidirect
121. #541Dic3semidirect
122. #540Dic3semidirect
227. C4⋊Q8C2×Dic3semidirect
228. C4⋊Q8C2×Dic3semidirect
229. #127C2×Dic3semidirect
233. #125C2×Dic3semidirect
235. #124C2×Dic3semidirect
247. C4×Q8C2×Dic3semidirect
251. C4×D4C2×Dic3semidirect
508. C3#21419semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#34552
Dic3×#1172
#822092
#718402
#718392
#718162
#718152
#717122
#717112
#716432
C4×#8372
#214193